Ionospheric Alfv6n resonator revisited' Feedback instability I V. Khruschev i •/[. Parrot 2 S. Senchenkov, 1

HAL Id: insu-03234728

https://hal-insu.archives-ouvertes.fr/insu-03234728

Submitted on 25 May 2021

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Ionospheric Alfv6n resonator revisited’ Feedback

instability I V. Khruschev i

•/[. Parrot 2 S. Senchenkov,

1

Oleg Pokhotelov, V. Khruschev, Michel Parrot, S. Senchenkov, V Pavlenko

To cite this version:

Oleg Pokhotelov, V. Khruschev, Michel Parrot, S. Senchenkov, V Pavlenko. Ionospheric Alfv6n

res-onator revisited’ Feedback instability I V. Khruschev i

_{•/[. Parrot 2 S. Senchenkov, 1. Journal of}

Geophysical Research Space Physics, American Geophysical Union/Wiley, 2001, 106, pp.25813-25824.

�insu-03234728�

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. All, PAGES 25,813-25,824, NOVEMBER 1, 2001

Ionospheric Alfv6n resonator revisited'

Feedback instability

I V. Khruschev

i •/[. Parrot 2 S. Senchenkov,

1

Oleg A. Pokhotelov,

,

,

V. P. Pavlenko 3

and

Abstract. The theory of ionospheric

Alfv6n resonator

(IAR) and IAR feedback

instability is reconsidered. Using a. simplified model of the topside ionosphere, we

have reanalyzed the physical properties of the IAR interaction with magnetospheric

convective flow. It is found tha, t in the absence of the convective flow the IAR

eigenmodes

exhibit a strong da,

mping due to the leakage

of the wa,ve

energy through

the resonator upper wall and Joule dissipation in the conductive ionosphere. It is found that maximum of the dissipation rate appears when the ionospheric conductivity approaches the "IAR wave conductivity" and becomes infinite. However, the presence of Hall dispersion, associated with the coupling of Alfv•n

wave modes with the compressional perturbations, reduces the infinite damping of

the IAR eigenmodes in this region and makes it dependent on the wavelength. The

increase in the convection electric field lea, ds to a. substantial modification of the

IAR eigenmode frequencies a. nd to reduction of the eigenmode damping rates. For

a given perpendicular wa,velength the position of inaximum damping rate shifts to

the region with lower ionospheric conductivity. When the convection electric field

approaches a certain critical va,lue, the resona,tor becomes unstable. This results

in the IAR feedback instability. A new type of the IAR feedba, ck instability with

the lowest threshold value of convection velocity is found. The physica,1 mecha, nism

of this instability is similar to the Cerenkov radiation in collisionless plasma,s. The

favorable conditions for the insta, bility onset are realized when the ionospheric

conductivity is low, i.e., for the nighttime conditions. We found that the lowest

value of the marginal electric field which is ca, pa, ble to trigger the feedback instability

turns out to be nearly twice smaller tha, n tha,t predicted by the previous analysis.

This effect may result in the decrease of the critical value of the electric field of

the magnetospheric convection tha,t is necessary for the forma, tion of the turbulent

Alfv•n boundary layer and appearance of the a, nomalous conductivity in the IAR

region.

1. Introduction

The ionospheric Alfv•n resonator (IAR) has been identified in the ground-based observations both in mid-

die [Polyakov and Rapoport, 1981; Bel•lc•ev et al., 1987,

1990] and in high latitudes [Belyaev ct al., 1999]. The

analysis of Freja satellite data [e.g., Grzcsiak, 2000] also

confirms the existence of this phel•omenon in the top-

side ionosphere. The IAR can be excited through a

•Institute of Physics of the Em'th, Moscow, Russia

2 Laboratoire de Physique et Chimie de l'Environnement,

number of natural and n•a.nn•ade ilnpacts. For exam- ple, it becomes unstable in the course of heating of the ionosphere by a powerful HF radio signal with a. mod- ulation frequency which lies in the range of IAR eigen-

frequencies (0.1- 10 Hz) [Blagovcshchenskaya st al.,

2001]. A comprehensive review of the artificial excita-

tion of Alfvdn waves in the IAR has been given recently

by Trakhtengertz et al. [2000], who discussed two prin-

cipal ways of such triggering. The first one assrunes the use of two spaced transmitter al•tennas which produce an ionospheric current being in resonance with the IAR eigenmodes. The second method is based on the change

Centre National de la Recherche Scient. ifique, OrlOans, [•h'ance in the macroscopic parameters of the ionosphere, such

3Department of Space and Plasma Physics, University of

Uppsala, Uppsala, Sweden.

Copyright 2001 by the American Geophysical Union.

Paper number 2000JA000450.

0148-0227/01/2000JA000450509.00

as conductivity in this region. The mechanism for the nonlinear excitation of the IAR by elves- a.nd/or sprites- produced lightning discharges has been discussed by

,5•ukhorukov and 5"tubbe [1997], who suggested that the

a, nom•lously large ULF transients observed in the up- per ionosphere on board the satellites above strong at-

25,814 POKHOTELOV ET AL.- IONOSPHERIC ALFV•N RESONATOR

mospheric weather systems [Fraser-Smith, 1993] are ex-

plained by excitation of the IAR eigenoscillations. A basic mechanism of the IAR excitation in high latitudes is connected with the resonant interaction of the magnetospheric convective flow with the conductive ionosphere. If the convective flow is properly phased with the ionosphere and the magnetosphere, part of the energy of the convective flow is transferred to the IAR eigenmodes. This effect, is known a.s the feedback in- stability, and it has been extensively studied by Lysak

[1986, 1988, 1993, 1991, 1999] and Tr'akhte•gcrtz a•'•d

Feldstei• [1981, 1984, 1987, 1991]. More general anal-

ysis of the IAR and the feedback instability has been

provided recently by Pokhotelo•' ctal. [2000]. In par-

ticular, the role of Hall divergent currents and, associ-

ated with thein, Hall dispersion of the IAR eigenmodes,

has been discussed. It was shown that the decelera.tion

of the Alfv•n wave phase velocity in the IAR due to

Hall dispersion may increase the ra.te of energy transfer from the convective flow to the 1AR eigenmodes and may overcome the dissipation rate due to Joule hea, ting and the leakage of energy through the resonator tipper boundary. The physical mechanisms of the feedback

instability has much in common with the (,erenkov ra.-

diation in collisionless plasmas [Pokhot½lov et al., 2000].

It should be noted that the role of divergent Hall currents in magnetosphere-ionosphere coupling was re-

cently emphasized by )'•shikawa a•'•.d Ito•aga [1996],

and this effect was extensively studied by Buch. c•'t and

Budnik [1997], Yoshikawa et al. [1999], and Yosh. ikawa

and Itonaga [2000]. Lysak [1999] has recently performed

numerical modeling of the magnetosphere-ionosphere

system which also confirms the importance of Hall dis- persion effects in the course of this interaction.

The physical reason for the IAR occurrence is due to

a strong increase in Alfv•n velocity with altitude, which

results in violation of WKB approximation and subse-

quent wave reflection from velocity gradients. The lat-

ter leads to formation of a resonance ca.vity in the top-

side ionosphere. The analysis of the feedback inst, a. bil-

ity provided by Lysak [1986, 1988, 1993, 1991], Tt-'okht-

e•'•gcrtz and Feldstei•'• [1981, 1984, 1987, 199]], and

Pokhotelov et el. [2000] was based on the assumption

that Alfv•n velocity in the ionosphere varies accord-

ing to a simple analytical exospheric profile

a•'•d Gveifinger, 1968], which may be considered as rep-

resentative. This makes it, possible t,o express the IAR

eigenfunctions in terms of Bessel functions and to obt, a.in simple analytical expressions for the eigenfrequencies a, nd the growth rates in two limiting cases of low and high ionospheric conductivity. However, in the most im- portant case of finite ionospheric conductivity such a.n approach uses a quite complicated numerical a. na. lysis

[e.g., Lysok, 1991] which does not, ofi, en allow the draw-

ing of a simple physical picture of the involved physical processes.

In this paper, using a simplified model for the Alfv•n velocity profile, we will extend the previous anal.x,'sis of

the IAR and the feedback instability for arbitrary val- ues of the ionospheric conductivity. We xvill analyze the properties of the IAR when it is disturbed by presence

of convective flow. In addition to the earlier studies we will show that most favorable conditions for the IAR

feedback instability are realized at low but finite iono- spheric conductivity. We will find a new IAR instability

which may arise at the value of the convection electric

field that is smaller than predicted earlier.

The paper is organized as follows' section 2 is de- voted to the derivation of a general dispersion relation for the IAR eigenmodes. The al•al.vsis of the IAR in the absence of the convective flow is given in section 3. The analysis of the IAR feedback instability is presented in

section 4. Our discussions and conclusions are found in

section 5.

2. IAR Dispersion Relation

In order to make our consideration as transparent as possible we consider a simplified model of a three- layer ionosphere shown in Figure 1. This model treats

the ionosphere as a height-integrated slab, neglecting

Hall and Pedersen conductivities above the ionospheric

height. A similar model has been used to calculate the propagation of Alfv•n waves in inhomogeneous media

by Malli,,ckrodt a,,d Carlso•'• [1978]. The ionosphere-

atmosphere interaction in our model is described by the

conductive slab with a two-dimensional tensor conduc-

tivity

•_ ( Ep

--•U

)

(1)

where E• and ZH correspond to height-integrated Ped-

ersen and Hall conductivities, respectively. In such a

model the IAR is localized in the homogeneous layer

with the Alfv&• velocity

defined

a.s

'vm•- B/(p,

op•)

•/'2,

where B is the magnitude of the ext, ernal magnetic field B,/u0 is the permeability of free spa.ce, and p•r is the IAR plasma mass density. The homogeneous layer above the

IAR with plasma density PM • P• represents the mag- netosphere. We assume also that in the unperturbed

state, there is a convective flow moving with the con-

stant velocity v0 - (E0 x 2)/B, wl•ere Eli is the elec-

tric field due to the magnetospheric. convection a.nd 2 is a. unit vector along B. As we shall see in wha.t fol- lows, this simplified model qualitatively describes all

basic features of the problem at, hand. In addition, it allows us to reveal additional physical features of the

IAR. which have not been discussed earlier.

Let us consider all perturbed qua.ntities to va.ry as

exp(-iwl), where w is the wave frequency. In the refer-

ence frame moving with the convection velocity v•, the

shear Alfve'n and compressional modes are described

by a set, of differential equations [cf. Pokhotclov et al.,

'2000]

POKHOTELOV ET AL.' IONOSPHERIC ALFVI•N RESONATOR 25,815 0 ,Az ... ,.1 ... • ... I ... A• //////////•// Em'th

Figure 1. Schematic diagram of the IAR model. The variation of perpendicular electric field 5Eñ with the altitude for the fundamental eigenmodes is shown. The case of low ionospheric conductivity is indicated by a solid curve, whereas the dashed curve corresponds to high conduc-

tivity.

2

+ -0, (3)

where

VA(Z)

-- B/(t*oP)

•/2 is the altitude-dependent

Alfvdn velocity,

p is the plasma

mass

density,

?), -

O/Oz,

and X;

'2 - 0• + V•.. The potentials

(I) and •P are

connected with the perturbed electric 8E_c and mag-

netic 8B_c fields by the following relations

5Eñ - -V'ñ (I) + iwW ñ x (4)

6Bñ - V_cA (5)

*Bz -

Here the parallel component of' the vector potential A and the scalar potential (I) are not, independent be- cause we consider that the field-aligned electric field in our plasma is zero; that is, E• - -½?•(I)+ i•A - O. Such an assumption is justified if the wavelengths of

the considered

perturbations

are nmch

larger than the

collisionless electron skin depth.

The solution

of equation (2) in the internal region

(IAR region)

has the form of a superposition

of down-

going and upgoing waves

(I) = A• exp(izw/vA•) + A• exp(--izw/vA•). (7)

In the external region (magnetospl•ere) we assume that only the outgoing wave remains, that is,

(]? -- C exp(iszw/vAl), (8)

where s - VAI/VAM • 1 is the ratio of the Alfv6n ve-

locities in the IAR and in the magnetosphere and

A2, and C are constant values.

According to Trakhte•gertz a,,d I•:lclstei• [1991], such

an approximation corresponds to the so-called radiation condition at infinity. The necessary matching of solu-

tions (7) and (8) at the tipper IAR. boundary, that is,

at z - L, requires the continuity of both the pot, ential and its first derivative. The necessary matching at the

25,816 POKHOTELOV ET AL.' IONOSPHERIC ALFVI•N RESONATOR

upper IAR wall gives

(VAI/co)Oz(I)

+ iOzp(I)

+ COOZHIII

+ B],.]-2[c•(p(kñ

x v0)•

(9)

c7 - - xp[i( - (lo)

where

:co

- wL/vAi. The terms

of the order

of •'2 in (9)

and (10) are neglected.

We consider the perpendicular wa.velength to be nmch

smaller

than both VAt/Co

and VAM/W.

Replacing

V]_ by

-k]_ (where

kñ is the perpendicula.r

wave

l•umber)

in

equation (3), we find that the compressional component

of the magnetic field, described by the potential q•, de-

cays as cr exp(-kñz) in both regions, tha,t is,

where •0 is a constant value.

These equations should be supplemented by two boun- dary conditions on the conductive slab (~ - 0). The ionosphere-atmosphere interaction is controlled by the height-integrated current flowing in the lower ionosphere. In our reference frame the conductive slab moves with the velocity -v0. Thus Ohm's law, integrated across

the slab, can be written a.s [cf. Pokhote:lo.'v _el al., 2000]

5Ji - EpSEi+EH2 x

where •J• denotes 6he change in the perpendicular height-integrated electric current and 5• is the pertur-

ba, tion of the ionospheric density which arises in •he

ionosphere due to the particle precil)itation and recom-

bina, tion processes. Hall •nd Pealersen conductivities

a,re a,ssumed to be uniform, and their perturbed values

5Ev •nd gEH are considered to scale a,s ,• 5.. This is a

re•sonable approach as long as the ionospheric temper- a,ture and 6he neu•rM densiW do not vary signifi(:a,nfiy because of the field-a,ligned currents or subsequenl, par-

ticle precipitation [Lgsak, 1991]. Expression (12)shows

6ha,6 6he presence of the convectiw• flow may substan- tia,lly modify 6he ionospheric current a, nd c•n signifi- ca, n61y change 6he properties of the lAR.

We a, ssume that the field-aligned current does not penetrate into the insulated a,tmosphere; that is, the

condition

Jll(-•z) - 0 is satisfied A•cordmg

to Am-

pbre's law the latter means that divergence of 6Jz pro- duces only the ionospheric field-a.ligned current, and

thus we have iki ß

5Ji - -ill on the upper

boundary

of the conductive slab. With the help of the para.llel

component of Ampare's law this condition reduces to

+OzH{kñ. v0)]-- -- 0. (14)

Here c•ip - E•/Ew and C•H -- EH/E.•,, are dimen- sionless Pedersen and Hall conductivities, respectively. Usually, in the ionosphere they are connected to each other. For example, for sunlight-produced conductivity, a• • 2a•, whereas for precipitation-produced conduc-

tivity, •/o,• • E •/8, where

E is the energy

of precip-

itating electrons in keV [e.g., Spiro et al., 1982]. This

ratio is 4.2 for 10 keV electrons and 2.7 for 5 keV. Large values of aH/a• of the order of 5 a.nd higher ma.y take

place in the auroral zone (e.g., B'r'ekke et al. [1974];

see also discussion provided by Y7).shikawa a•d Ito•aga

[1996]).

The change in •he ionospheric density is con[rolled by •he con•inui[y equa[ion, which in our reference frame

takes the form [Lvsak, 1991]

-iwgn

+ iki ß

(ngv•)=

7 c•zJ

'

where 5v• - (dEñ x •)/B, e is the magnitude of

electronic charge, •z' is the depth of the conductive slab, 7 represents the number of additional electron-ion

pairs created per incident electron, and v is an efibctive

recombination kequency.

The solution of equation (15) gives

n w + iv

Here r• - (•Z/Spi)7 -1 is the dimensionless

parame[er,

which con[rols the elec[ron precipitation, 5•,i - C/•pi

is the collisionless ion skin depth, c is the speed of

light,

•pi -- (ne2/Somi)

1/2 is the ion plasma

fi'equency,

and mi is [he ion mass. For the actual ionospheric

condi[ions, 7 is nearly a linear timerñon of [he incom- ing elec[ron energy. It is about 100 IBr 10 keV dec-

trons [Bothwell

et al., 1984]. According

to Lusak

[1991]

• • • 5pi • 10-- 30 knl. Therefore (•z is a. sraM1 para.me-

[er of [he order of 10 -•- 10 -2 A typical recombina.[ion

fi'equency

for [he ionospheric

density

of n • 10

• m -3

a,nd

recombina[ion

coefficient,

R - v/2n • 10

-13 m 3

s -• is v• 10 -2 s -•

Combining (14) a.nd (16), we obtain

(1 •99 VAI

_•o+iv

_{) •o z}

0 (I)+iozp(I)+CoCrH

(1

-

where

Co + il/

•--• it, t;AI

kñ .SJñ: (;:(I), (13)

where Ew - 1/ItOVAI is the "IAR wa,ve conductivity."

Dotting 6J z, given by (12), with kñ and using (13),

gives

with

POKHOTELOV ET AL.' IONOSPHERIC ALFVt•N RESONATOR 25,817 If ß is properly expressed through (I), equation (17),

together with solutions (7)-(11), describes the disper-

sion relation for the IAR eigenfrequencies. The neces-

sary connection can be obtained by applying (V x), t,o

Ampbxe's

law, which gives

[cf. Pokhotelov

et ol., 2000]

where 5jñ is the change in the perpendicular iono-

spheric current.

Integration of (19) gives

O:N(O)-O•

•(-Az)-k•_Az•I•(O)

- -i/,,,/,:7:'(kñ

xSJ

('20)

On one hand, according to (11), •i• [alls off exponentially above the conductive slab, i.e., •I• ..x exp(-kñ z'). On the other hand, in the neutral atmosphere (-d < z< .Az),

• varies a.s [e.g., Yoshikawa a•d Ilo•'•.uga, 1996]

ß _

(: + '2d)] (21)

1 - exp(-2kñ d)

where C is a constant value.

Expression (21) is valid if ß vanishes on the surface

z : -d. Such simplification assumes that the solid Earth is considered to be a perfect conductor. In reality, magnetometer chains above the solid ground do register large-scMe variations of external origin in •, suggesting

that • -y= 0 [Buchert a•d B'u&•ik. 1!.)97]. Thus, st. rictly

speaking, our assumption of a conducting Earth is sat-

isfied above the sea. and (:lose l.o it,. Otherwise, the

effects due to the finite Earth's conductivity should be

taken into consideration, and (21) should be replaced by a more complex expression. (-'onsiderat. ion of this effect is, however, out of the scope of the present pa- per. We note that the limit of nonconducting Earth

was discussed by Buchert and Bud.nik [1997]. In the

limiting case kñd >> 1 the compressional mode falls off

as exp(-kñ I z I) on both sides of the ionospheric sheet

[Pokhotelov et al., 2000].

With the help of (21) we find

1_ - exp(-2kñd)

•[t,: k_LVAI

kñvAI(W

+ i•-')]

-- kñ

VAI

(24)

where

]? ñ 'u ( ) o.: c

A•: -•

sin[g - al'('l,all(Op/O:H)]. (25)

(11

Equations (17)and (24), together wit, h (7)-(1:2), fbrm

a closed system of equations which describes the dis- persion equation for the IAR eigenmodes in the pres- ence of the convective flow. They generalize the ('orre-

sponding equations of Pokhotelo.'v •:t al. [2000] by in-

clusion of finite k i•z' and kid effect. s. In a special

case when k•d 4< 1 (which also assumes k•z' <4 1)

we have g - (kzd) -• >> 1. This sit.

uation was ana-

lyzed 1oy Pol:qakov u•d Rupopo;'l [1981]. We not. e that

since for the real conditions d • 10 2 kin, such an ap-

proximation is relevant only for t. he perloendicular wave

lengths lz 55 2•d • 600 kin. Such waves do nor sat-

isfy the •;erenkov

resonant

condition

that,

is necessary

for the IAR excit, at, ion by magnetospheric conveer, ire

flow [e.g., Pokhotelov ctal., 2000], thus another lim-

iting case for which k zd >> 1 seems t.o be more rele-

vant for our study. Such waves are effectively trapped

in the IAR and can accumulate a su•cient amount of

energy to influence the aurora [T•'akhte',ge•'tz a•cl Feld-

stein, 1987]. In this limiting case coth(kzd)• ] and

g • l+kz&z+exp(-kzAz). We note l. ha.t the analysis

of Pokhotelo'v et al. [2000] corresponds t,o the so-called

zero-Az approximation, when kz.Az 5 ] and •;. - 2. Since for the real ionospheric condit. ions &z' can be of the order of one tenth of a kilometer, another limiting case, when kz&z >> 1 and a:- /;'z•z, may be<'ome more relevant for the interpretation of observations.

3. IAR in Absence of Convective Flow'

Role of Hall Dispersion

Let, us consider the case when t, he convection electric

field vanishes, that is, E0 - 0. In addition, we neglect the corrections due to Hall dispersion. In this limiting

(rase, equation (17) reduces t,o L,so/,: [1991] type

..o Ckñ exp(-kñAz)coth(kñd), (22)

xvhere we assumed that the depth of the conductive sial) is much smaller than its altitude above t, he ground, i.e.,

Az<<d.

Using (11) and (22), we find

ß = i/_t0ki3(kñ

x 6az)z

(23)

where • - 1 + kñAz + exp(-kzAz')cot, h(kñd). With

the help of (12) from (23) we fina,lly

obt,

ain

(VAi/C0)Oz(I) q- iOp(I) -- 0,

Substitution of (7)-(10) into (26) gives

(26)

l+a•p

- (27)

exp2iz0

(1 + 2•)l_ op

Decomposing the dimensionless frequency in equation

(27) into the real and imaginary parts, .•0- tl + i')', we

get

25,818 POKHOTELOV ET AL.' IONOSPHERIC ALFVI•N RESONATOR

where O• - arg[(1 + c•p)/(1- ,•,•)] and k is integer,

that is, k -0, 1, 2, .... From equation (28) ;ve obtain

(29)

r/- 0 _< < 1, (30)

(2k + 1)

•/- ••r

1 < c•p

< oo, (31)

The following comments are in order. The first term

on the right in (29) describes

the IAR damping

due

to the leakage of the wave energy through the res-

onator upper wall, whereas the second term corresponds

to losses due to the ionosphere Jotlie heating. When

c•p -• 1, that is, when the ionosphere and the lnagne-

tosphere are optimally matched, the damping becomes

infinite. Moreover, when ap - l, the mode frequencies undergo a finite jump. We term the region with sharp

frequency variation as a transition region [cf. Yoshikawa

et al., 1999]. It should be mentioned that here Hall dis-

persion, which has been neglected in equation (27), may

play an essential role. We will return to this problem

in the end of this section.

The plots of dimensional frequency and damping rate

as a function of ap are shown in Figure 2, using solid

curves. Figure 2 shows that through the transition region, the kth harmonic for the insulator-like iono-

sphere is connected with the (k + 1)th harmonic for

the conductor-like ionosphere. One of the peculiar fea-

tures of the IAR eigenmodes

shown

in Figure 2 (top)

is related to the fact. that the counterpart for the fun-

damental harmonic (k = 0) for the conductor-like iono-

sphere (ap > 1) corresponds to a. zero-frequency mode

in the insulator-like

ionosphere

(ap < 1). This vari-

ation of mode frequencies and the damping factor is

similar to that discussed

by Newto• et al. [1978]

and

Miura et al. [1982] for the localized toroidal oscilla-

tion under the dipole magnetic field model. Recently,

Yoshikawa et al. [1999] provided a. physical interpreta-

tion for this connection. According to this paper the ex-

istence of interconnected harmonics arises from a type

of conservation of parity in that the spatial structure

of the eigenmode along the field line is either symmet-

ric or antisymmetric

in nature. Conservation

of parity

assumes the keeping of a symmetric or antisymmetric

nature of field line oscillation even when the geometrical

changes of oscillation have occurred by the parametric

change in ionospheric conductivity.

We note that the values of the eigenfrequencies, ob-

tained in our model, are different from those obtained

in a model with smooth exospheric profile of the Alfv•n

velocity

[e.g., Lysak,

1991; Trakh.

te•,gertz

a•d Feldstein,

1991]. We recall that in such a model,

variation

of the

Alfvdn velocity with the altitude is described by a sirn-

ple expression [Greifi•#er a•d Gr'eifi•#er, 1968]

5 i .-

1

....

.,'

/'

Figure 2. Variation of the frequency and the damI>

ing rate for the IAR eigenmodes a.s the function of ap

in the absence of convective flow (s - 0.01, u - 0.01,

k•_d- 60, d- 100 kin,/kz - 30 km and L - 1000

kin).

(Top) Solid curve shows the variation of dimensionless

eigenfrequency as a function of the dimensionless Ped- ersen conductivity in the absence of Hall dispersion. Dashed curve corresponds to the case when Hall dis-

persion

effect.

is included

(all-

O'p). (Botton•)Same

as top but for the damping rate.

V•I (Z)

(:)- +

'

where, as in our model, e • 1 defines the ratio of the

Alfv•n velocity VA• in the ionosphere to that in the outer

magnetosphere YAM so that VA,/YAM • e.

According

to Lysak [1991] and Trakhte•gertz

a•d

[991], for

<<

high

>>

ionospheric

conductivity

the IAR dimensionless

eigen-

frequencies are defined by the roots of the Bessel func-

tions of the first and zero order, respectively. The first

three roots

of equation

J• (V) = 0 are 0, 3.8, and 7. They

are close to 0, •r, and 2•r and describe the dimension- less eigenfrequencies of the IAR in our model in the low

conductivity case. Similarly, for highly conductive iono-

sphere

we have J0(•]) = 0, and the corresponding

roots

are 2.4 and 5.5 which are not essentially different from

•r/2 and 3•r/2. Thus our simplified

model

qualitatively

POKHOTELOV ET AL- IONOSPHERIC ALFVI•N RESONATOR 25,819

According

to (29) the resonator

undergoes

a weak

damping in the two limiting cases of low and high iono-

spheric conductivities. Making t. he corresponding ex-

pansion, from (29) we get

7=-s-c•P

ap<<l,

(33)

>>

(34)

If one neglects this damping, then the IAR eigenfunc-

tions in these two limiting cases will take the form

• : 2A•[cos(zco/vA•)

- • exp(-izw/'v•x•)]

(35)

•: 2iA•[sin(zco/vAi)-

izexp(--iz"oO/VAi)] o? • 1,

which correspond to a superposition of a standing wave

and a wave reflected from the upper boundary of the

IAR wall. When • << 1 and o• >> 1, the resonator

eigenmodes are as shown in Figure 1, with bold and

bold dashed curves, respectively.

The inclusion of HM1 dispersion removes a disconti- nuity in the variation of the IAR damping rate as well as the jmnp in the variation of eigenfrequency. More-

over, the presence of Hall dispersion replaces the zero-

frequency mode by a finite-frequency mode. To demon-

strate this, we generalize the dispersion equation for the

IAR eigenfrequencies by considering this effect. Then

equation (27) can be written as

exp(2iz0)

- (1

+ 2•)

l-6,•'

(37)

where

-

-

' (38)

The contribution of the second term on the right into

equation (37) is always small, except for the case when

• 2 1. The presence of this additional term will smooth the variation of the frequency and the dalnp- ing rate in the transition region which now scales a.s

• ln(nkzL) at •p: 1. This is shown in Figure 2 (up-

per and lower panels) by a dashed curve. It is worth

lnentioning that a similar effect of disappearance of sin- gularity at •p: 1 due to Hall correction in the expres- sion for the wave reflection coefficient can be revealed

by Polyakov and Rapoport [1981] analysis (e.g., equa-

tion (11)). However, this expression was obtained in

the limiting case k zd << 1 when IAR eigenmodes do not effectively interact with the magnetospheric con-

vective flow. Fine peculiarities of the change in mode

frequency and damping rate becolne more clear if we

make an expansion of equation (37) in two limiting

cases of small and high ionospheric conductivities. For

low ionospheric conductivity (•p << 1) from (37) we ob-

tain, - 0 for k - 0 and, - •(1-a,•/nkmL) for k - 1.

1 60 1 59- 1 58 1 57 1 56 1 55 1 10 100 O(p

Figure 3. Variation

of the dimensionless

mode

fre-

quency

in the vicinity of 'q •_ •/2 for different

values

of C•H/c•p.

Solid

curve

corresponds

to C•H/C•p

: 12,

dashed curve corresponds to OHio,,: 1.75. Other pa-

rameters are the same as in Figure 2.

This agrees with Pokhotelov et ctl. [2000], who demon-

stra, ted that the IAR eigenfrequency at k: 1 undergoes a red frequency shift induced by Hall dispersion. The zero-frequency solution is not influenced by Hall disper- sion in this approximation, whereas the damping rate

becomes

smaller

7 -- -(e + o:•)/(1 + o"•/a;kzL). In

the opposite limiting case, a,:• >> [, the correspond- ing expansion of equation (37) fbr the lowest mode

(k - 1) gives

•1- •/2 +2a•k•L/•(o,•

+ o,•) '2 and

7 - -•-

ap/(c• + a•). Thus the mode frequency

exhibits a blue frequency shift, due to Hall dispersion, and the damping rate is substantially modified by the presence of Hall conductivity. The latter is connected with fact that a divergent Hall current becomes large for high conducting ionosphere, and the divergent Ped-

ersen current is shielded [cf. Ybshikawa and Itonaga,

2000]. The change in fundamental mode frequency for the finite values of a• and different ratios of r•H/r•p is shown in Figure 3. One can see that Hall effects lead to the appearance of a maximum in the variation of the mode frequency. A similar effect was reported by

5½shikawa a•d Ito•aga [1996] in the course of the anal-

ysis of wave reflection a.t the ionosphere. Since they have analyzed much longer wavelengths, this maximum is more pronounced in their study than in our case, when Hall corrections are small.

It is worth mentioning that rigorous consideration of the limiting case of perfectly conducting ionosphere

(a• >> 1) requires

a special

consideration

which is out

of the scope of the present study. We note that since the IAR eigenperiods are usually smaller than several sec- onds, the approximation of the thin sheet conductive layer may not be satisfied in this limit. The validity of this approximation requires that, the depth of con- ductive layer Az must be smaller than the skin depth

• : (2/p0•e•) 1/2, where

e• stands

for Pedersen

conductivity and • is the IAR eigenfrequency. This

inequality

can be rewritten

as Az << (2AzL/•,,o,•) •/2

25,820 POKHOTELOV ET AL.' IONOSPHERIC ALFV]•N RESONATOR

damental mode •'?,•. • rr/2). It is evident from this es-

timation that the thin sheet approximation, adopted

in our analysis, may be violated in the conductor-like ionosphere for very large values of o,'p.

4. IAR Modification Induced by

Convective Flow: Feedback Instability

Let us now consider how the presence of convective

flow modifies the physica,1 properties of the IAR eigen-

modes. For this aim, equations (17) and (24)have

been solved numerically. As an exa.mple, the change in

the dimensionless frequency of the lowest (fundamental)

IAR eigenfrequency for different va.lues of rr - rrvm•x

- r, m•xL/ and

is shown in Figure 4. Here C%m•x = -•½ VAI

99 -- 99m•x stands for the a, ngle of propa.gation, calcu-

lated numerically, which yields the ma.ximum value of

growth/damping rate. It, is clea.rly seen tha. t with the

increase in the convection electric field the major mod- ification of the mode appears basica.lly at o:p • 1, that is, when the ionospheric conductivit, y is low. The mod- ification results in two effects. On one hand, the t, ran-

sition region shifts toward the lower ionospheric con- ductivity. On the other hand, the fundamental eigen- mode frequency increases in value. The damping of the IAR mode becomes weaker. Figure 5 shows the change in the damping rate of the fi•ndamental mode. With the increase in the convection electric field the ma.xin•um value of the damping ra.te decreases and the

position of a minimum shins to the lower conductiv-

ity. Eventually, the growth rate becomes positive when

rr > zr/2. The condition rr • zr/2 corresponds to the

marginal stability of our system. in this limiting case

equettions (17) and (24) possess a.n a,nalyticed solution.

We note that for marginal sta10ility, cr --• rr/2, the value

ofa vanishes, S• --• 0, and thus the cor-

responding terms in equation (17) a.re small [Pokhotelov

et al., 2000]. Taking into account these considerations,

fi'om (17) we obtain -. -.. 03 O0 -'

•. : , .""

...

:7-

...

Figure 5. Same a,s Figure '4 but for the damp-

c•p+

(1-c;• ) 1-(l-2s)eui•'ø

-

+ i7 )- o,

(39)

where, contrary to section 2, the recombination fie-

quency • is dimensionless, that is normalized to 'vmi/L.

Equation (39) can be further simplified. Similar to

Pokhotelov et al. [2000], it can be shown that the last

term on the left, which describes the corrections due to Hall dispersion, leads to a sma, ll increase in the wave growth rate. Neglecting this effect,, we finally obtain

... 0 (40)

•0 + i• 1 + (1 - '2..z)e •'i•o '

Let us search tbr the solution of (40) near the lowest

IAR resonance, that is, in the vicinity of the solution of

equation

1 + (1 - 2s)e

=i•ø

= 0. The latter equation

has

- '* - rr/2- i•.

the lowest weakly damping root .• x 0

Introducing Ax0 - x0 - z•, from (40) we obtain

-- •:4).25 .... •-1 .!5 3.0 "'.:. - ,,-0.5 •,,:-2 *

25

,,,,"•

...

----•

125

Pigure 4. Variation of the IAR fundamental eigenDe- quency as the function of o,p for different, values of •.

Here •H/•P -- 1.78. Other parameters are the same as

in Figure •.

(xo - x;)(1 - ict,

px; -+-

•,ozp)- (cr

v, - x;)q-- iv, - O. (41)

When splitting x0 into real and imagina, ry parts, we find for the respective dimensionless fi'equency and growth rate

•o:.,,:,

(s _ ,..,)

(42)

r/- cr•

2

'

•ro•p •r

(43) From the first term on the right in (43) one imme-

diately recognizes the damping due to recombination processes. We note that the damping due to the energy leakage through the upper IAR wall is exactly coun- terbalanced by the effect of accumulation of energy in the resonator due to the wave reflection from the IAR

POKHOTELOV

ET AL.. IONOSPHERIC',

ALFVt•N RESONATOR

Using (18) we find that instability

('• >_ 0) etrises

for

the angles of propagation

a,p V cr

cos(p- arctan ) <_

---,

(44)

[

]

__ OZIVAIlr

I d- --

(45)

Such angles

exist if v0 >_ vg", which defines

the in-

stetbility threshold. When v0-

with [cf. Pokhotelov

et al., 2000]

92m•x

"'"'

-]-71'

d-

a,rcta,l•

c•

?

(46)

Substituting

92

- 92m•x

+ A92 into equa.tion

(41) etnd

using

the expansion

cos

A92

• 1- (A•)'-'/2, one

finally

obtains the expression for the instability growth ra.te

Considering v0 • vg • stability we ha,ve

, fi'om (47) [br the margina.1 in-

wit,h the Lysak [1991] model, which uses a, smooth exo-

spheric profile of the Alfv•n velocit,y, a. rises in the values

of the IAR eigenfrequencies. Hmvever, our model a.llows

us to provide a deeper insight into the physics of this

phenomenon. In particular, it was delnonstretted tha. t

Ha, ll dispersion pletys a. key role in the IAR dalnping when Pedersen conductivity a. pproaches the "IAR xva.ve conductivity." The inclusion of this effect into consid- eration removes inconsistency of the model related to infinite damping at a,e = 1. The dalnping ra.te becolnes

dependent on the wavelength and scales as c< ln(gkiL)

in this region. The cornerstone of our analysis is that a new IAR instetbility with the lowest t.hreshold has been

reveetled which wa.s not, studied in the course of t,he ear-

lier etnMysis.

It is worth ment,ioning that the IAR instetbility at

•1: •r/2 in the low conductive ionosphere directly fol-

lows front the model which uses a smoot. h profile of t, he

Alfvdn velocity. Let us show that. a silnilar solut, ion for

the feedbetck inst,ability exists in the Lysak [1991] betsic

equation. For simplicity, let us neglect. the correct, ions due to recombination and leakage out of the resona.tor

upper boundetry and consider v - • - 0. Not. e that

these effects should just provide t, he damping. Then

from Lysak [1991] we get

7-- • c•? 1+

-

1

.

(48)

The growing

waves

propagate

in the cone

defined

by

and centered around gm•x.

(Jerenkov

This scenario is similar to the radiation in

collisionless

plasma [cf. Pokhotelo'v

et al., 2000]. For

the marginM instability (v0 • v•.5") the growing waves

propagate

at an etngle

g- •max defined

10y

(46).

For vii - 500 km/s etnd L - 1000 km et t, ypical

dimensionless recombination fi'equency is of the order

of 5 x 10 -a and the nighttime conductMty •e • 0.1

[e.g., Lysak, 1991]. Thus v << (_•e << 1 and the re-

combination processes do not suppress the insta, bility

during

the night. Substituting

•x - 10

-• ' 10

-• and

•c • 0.2 into (45) we estimette

the criticM velocity

as

1.2s (10-' ß

xz

km, L • 1000 km and vii - 500 km/s this gives

v • • (60 ' 600)m/s. For B - 0.5 gauss

this corre-

sponds to the convection electric field of the order of

3 ' 30 mV/m, which

is typicM

for the polar ionosphere.

5. Discussion and Conclusions

It is shown that the IAR and the feedback instetbil-

ity can be qualitatively described in the framework of a

simplified model in which the Alfv&• velocity undergoes

a final jump at the upper boundetry. The only difference

(50)

In the low conductivity limit, that is, when av << 1, this equat,ion yields two solutions. The first one cor- responds to the case when solution to this equat, ion is

close to the roots of the Bessel function of the first order,

i.e., when J•(.•l•,,,) = 0. These roots are related to the

eigenmodes of the undisturbed resonettor in the lilnit of

low ionospheric conductivity. The lowest, nonzero root is equal to •1•0 = 3.8, which in our model corresponds to t/= •r. The instability growth rat, e in this ca.se wets

evMuetted by Lysak [1991] a. nd reads

7L -- 2-•/2(øZP•l•ll)

•/2

(51)

The necessa, ry threshold velocit3 .• for the inst.etbilit, y

onset was given by Pokhotelov et al. [2000] and etSSulnes

the form

v•_ a'•rVA•rV•o

(52)

kzLac,

However, ets it was shown etbove, equa.tion (50) yields

etnother solution locMized in the vicinity of roots of the

zero-order Bessel function, tha.t is, when Jo(xo) •- 0

etnd ere •_ •/0,,. This is reletted t,o the IAR eigenmodes

influenced by the presence of convective flow. The low- est instetbility threshold corresponds t,o •/00 = 2.4. We

recM1 that in et simplified model adopted in our study

this root is •r/2 (see Figure 4). Similar t.o section 4, the

expetnsion of (50) gives

ß S '

25,822 POKHOTELOV ET AL. IONO,_PHERIC ALFVI•N RESONATOR from which one readily obta.ins

with

7P - a,p •160 1 (54)

V 0 • •

kñLo,'(,

Expressions (54) m•d (55) are ident, ica.• t,o (48) a. nd

(45) if one replaces •100 by •/2 a.nd neglects the recom-

bination frequency v. Let us coinpare the x,a.lues of instability growth rates near •'1 2 •1•0 = 3.8 a.nd •1 = 2.4.

According to (51) m•d (54) this ratio is

72 Z 0.2a•

•/2

(56)

For typica,1 nighttime Pealersen conductivity a,p m 0.1

both these instabilities have the same order of va,lue,

i.e., 7L m 9'P. With •he increase in the ionospheric con- ductivity the instability growth ra,tes increase. Figure 6 shows the dependence of the instability growth ra,l,e a,s the fimction of ap in the vicinity of • m •/2. It is

clea,rly seen that it approaches the ma,ximmn va,lue

low but finite ionospheric conductivity. From compa, ri-

son of expressions (a0) a,nd(53) we conclude that, in-

s•a,bility threshold at •he frequency q m •100 is 1.6 i, imes lower than that at •1 m tl•0. Therefore it is reasonable t,o

assume that this instability can 1)e considered the most probable candidate for the IAR excitation.

Thus our analysis shows that the most, fa,vorable con- ditions are realized when the ionospheric conductivity

is low. •e no•e •ha,t this occurs ba, sica,lly during the nighttime conditions. In this ca,se the dissipation is small, a, nd the electric field of magnetospheric convec-

tion can easily penetrate the conductive Ma, b. In such

conditions the convection flow moves rela, tively freely

through

•he ionosphere,

losing

its energy

ba,sica,lly

due

to Cerenkov radiation in the IA•. These results agree

with •ewell el al. [1996] analysis of data collected from

the DMSP satellite showing that, the most intense auto-

002 • ... • ... • ... • ...

-- • 1

qq

?

...

,--

Figure 6. Variation of the growth rate of the funda- mental mode in the vicinity of •/• •r/2. Or, her param- eters are t, he same as in Figures 4 and 5.

03- 00 .... •=.().-s ½r 4.0 -06 • 12• .... • 6.o - - - •2 0 -09 .... I .... i ''' ' ' i .... I .... 0 20 40 60 80 00 O• H

Figure 7. Variation of the IAR, dalnping/gro•vth rate

a,s the function of aH for different, va,lues of •r. Here

a,? : 2, while other pa,rameters are the same as in Figures 2-6.

ral arcs (discrete aurora), which are usua,lly attributed

to s•nall-scale Alfvdnic structures [e.g., ,S'tasicwicz ½t

al., 2000], appear preferen[ia.lly in t, he xveakly conduc[-

ing ionosphere when st, tong electron precipita[ion is ob- served.

In section 4, for simpliciW, we ha.re neglected the

corrections due [o Hall dispersion effects which do noC

substan[ia, lly •nodify [he va, lues of the damping/grow[h

rates. However, a,s •vas mentioned by ¾bshika'wa a•d Itonaga [1996], their contribution becolnes nonva, nish-

ing for ra,ther la. rge va, lues of O'H a.t fixed a,p. Figure 7

represents an example of the variation of the instability

dmnping/growth rate as a funct, ion of O'H in the pres-

ence of Hall dispersion effects. One ca,n see tha,t these effects ma,y lead to the appearance of a. ma,ximum when

the ratio alllap is su•ciently large. Moreover, the in-

clusion of this effect into considera, tion is clearly ilnpor-

ta. nt for the ground obserwtions, since the direct lna, g-

netic signatures of the Pedersen a,nd field-aligned cur-

rent structures tend to cancel one a.nother [e.g., Ltlsak,

1999].

An important effect that wa, s not considered in our analysis refers to the electromagnetic stra, tifica. tion of the magnetospheric convection that arises a,s the result of the feedback instability. The detailed scenario of such process has been described by Tr'akhtcn. gertz and Feld-

stein [1981, 1984, 1987]. Moreover, the relevant non-

linear effects involved in the subsequent evoh•tion of the instability would lead to the forma.tion of a, turbu-

lent Alfvdn boundary layer (TABL) and appearance of

the anomalous conductivity in the IAR region [Trakht-

cngertz a•d Feldstein, 1984, 1991]. However, the de-

scription of these effects is out of the scope of the present

study. We should only mention that basic dispersion

equation (50) for the IAR eigenmodes yields a,n addi-

tional unstable mode. It starts to grow at the value

of the convection

velocity

defined

by expression

(55),

i.e., at the value of the convection electric field which is

POKHOTELOV

ET AL' IONOSPHERIC

ALFV•N RESONATOR

Thus the formation of TABL and associated phenomena

arise at more moderate ionospheric conditions.

The intention of the present approach is to provide

deeper

insight

into the physics

of the IAR and the feed-

back instability. Hence this paper can be considered as

an extension of our previous approach to the study of

the IAR [Pokhotelov

et al., 2000], which was bounded

by consideration

of low and high ionospheric

conductiv-

ity. The results

of our study

might

be useful

for a better

understanding

of the IAR properties,

as well as for the

interpretation

of recent satellite

observations

by Freja

and Fast satellites [e.g., Stasiewicz and Potemra, 1998;

Ergun et al., 1998; Stasiewicz

et al., 2000; Grzesiak,

000].

Acknowledgments. We are grateful

to Kalevi Mur-

sula, Jorma Kangas,

and Tillman BSsinger

for fruitful

discussions. One of the authors (O.A.P.) is grateful to

R. Z. Sagdeev

for valuable

comments.

The authors

ac-

knowledge

financial

support

from the Commission

of the

European

Union (grants

INTAS-96-2064

and INTAS-

99-0335). O.A.P. acknowledges

support

from ISTC

through

research

grant 1121-98

and wishes

to thank

the French Minist•re de la Recherche et de la Technolo-

gie for hospitality

at LPCE, where

this work was com-

pleted. V.P.P. acknowledges

support

from the Swedish

Natural Science Research Council (NFR) through grant

Janet G. Luhmann thanks Akimasa Yoshikawa, and another referee for their assistance in evaluating this paper.

References

Belyaev, P. P., S. V. Polyakov, V. O. Rapoport, and V.

netic propagation in the ionospheric waveguide, Jo Geo- phys. Res., 73, 7473, 1968.

Grzesiak, M., Ionospheric Alfvdn resonator as seen by Freja satellite, Geophys. Res. Left., 27, 923, 2000.

Lysak, R. L., Coupling of the dynamic ionosphere to auroral flux tubes, J. Geophys. Res., 91, 7047, 1986.

Lysak, R. L., Theory of auroral zone PiB pulsation spectra, J. Geophys. Res., 93, 5942, 1988.

Lysak, R. L., Feedback instability of the ionospheric reso- nant cavity, J. Geophys. Res., 96, 1553, 1991.

Lysak, R. L., Generalized model of the ionospheric Alfv•n resonator, in Auroral Plasma Dynamics, Geophys. Monogr. Ser., vol. 80., edited by R. L. Lysak, p. 121, AGU, Wash- ington, D.C., 1993.

Lysak, R. L., Propagation of Alfv•n waves through the iono-

sphere: Dependence on ionospheric parameters, J. Geo- phys. Res., l O d, 10,017, 1999.

Mallinckrodt, A. J., and C. W. Carlson, Relations between transverse electric fields and field-aligned currents, J. Geo- phys. Res., 83, 1426, 1978.

Miura, A., S. Ohtsuk_a, and T. Tamao, Coupling instability

of the shear Alfvdn wave in the magnetosphere with the ionospheric ion drift wave, 2, Numerical analysis, J. Geo- phys. Res., 87, 843, 1982.

Newell, P.T., C.-I. Meng, and K. M. Lyons, Suppression of discrete aurora by sunlight, Nature, 381,766, 1996. Newton, R. S., D. J. Southwood, and W. J. Hughes, Damp-

ing of geomagnetic pulsations by the ionosphere, Planet. Space Sci., 26, 201, 1978.

Pokhotelov, O. A., D. Pokhotelov, A. Streltsov, V. Khr- uschev, and M. Parrot, Dispersive ionospheric resonator,

J. Geophys. Res., 105, 7737, 2000.

Polyakov, S. V., and V. O. Rapoport, The ionospheric Alfv•n resonator, Geomagn. Aeron., 21, 816, 1981. Rothwell, P. L., M. B. Silevitch, and L. P. Block, A model for

propagation of the westward traveling surge, J. Geophys. Res., 89, 8941, 1984.

Spiro, R. W., P. H. Reiff, and L. J. Maher, Precipitating

electron energy flux and auroral zone conductances: An empirical model, J. Geophys. Res., 87, 8215, 1982. Stasiewicz, K., and T. Potemra, Multiscale current struc-

tures observed by Freja, J. Geophys. Res., 103, 4315,

Y. Trakhtengertz, Discovery of the resonance spectrum 1998.

structure of atmospheric electromagnetic noise background Stasiewicz, K., et al., Small scale Alfv•nic structure in the in the range of short-period geomagnetic pulsations, Dokl.

Akad. Nauk SSSR, 297, 840, 1987.

Belyaev, P. P., S. V. Polyakov, V. O. Rapoport, and V. Y. Trakhtengertz, The ionospheric Alfvdn resonator, J. At-

mos. Terr. Phys., 52, 781, 1990.

Belyaev, P. P., T. BSsinger, S. V. Isaev, and J. Kangas,

First evidence at high latitudes for the ionospheric Alfvdn

resonator, J. Geophys. Res., log, 4305, 1999.

Blagoveshchenskaya, N. F., V. A. Kornienko, T. D. Borisova, B. Thide, M. J. K osch, M. T. Rietveld, E. V. Mishin, R. Y.

Luk'yanova, and O. A. Troshichev, Ionospheric HF pump

wave triggering of local auroral activation, J. Geophys. Res., (in press), 2001.

Brekke, A., J. R. Doupnik, and P.M. Banks, Incoherent

scatter measurements of E region conductivities and cur-

rents in the auroral zone, J. Geophys. Res., 79, 3773, 1974.

Buchert, S.C., and F. Budnik, Field-aligned current distri- butions generated by a divergent Hall current, Geophys.

Res. Left., 24{, 297, 1997.

Ergun, R. E., et al., Fast satellite observations of electric

field structures in auroral zone, Geophys. Res. Left., 25,

2025, 1998.

Fraser-Smith, A. C., ULF magnetic fields generated by dec- trical storms and their significance to geomagnetic pulsa- tion generation, Geophys. Res. Left., 20, 467, 1993.

Greifinger, C., and P.S. Greifinger, Theory of hydromag-

aurora, Space Sci. Rev., 92, 423, 2000.

Sukhorukov, A. I., and P. Stubbe, Excitation of the iono-

spheric resonator by strong lightning discharges, Geophys.

Res. Left., 24, 829, 1997.

Trakhtengertz, V. Y., and A. Y. Feldstein, Effect of the nonuniform Alfvdn velocity profile on stratification of magnetospheric convection, Geomagn. Aeron., 21, 711,

1981.

Trakhtengertz, V. Y., and A. Y. Feldstein, Quiet auro- ral arcs: Ionosphere effect of magnetospheric convection stratification, Planet. Space Sci., $2, 127, 1984.

Trakhtengertz, V. Y., and A. Y. Feldstein, About excita- tion of small-scale electromagnetic perturbations in iono- spheric Alfv•n resonator, Geomagn. Aeron., 27, 315,

1987.

Trakhtengertz, V. Y., and A. Y. Feldstein, Turbulent Alfvdn boundary layer in the polar ionosphere, 1, Excitation con- ditions and energetics, J. Geophys. Res., 96, 19,363, 1991. Trakhtengertz, V. Y., P. P. Belyaev, S. V. Polyakov, A. G. Demekhov, and T. BSsinger, Excitation of Alfvdn waves and vortices in the ionospheric Alfvdn resonator by mod- ulated powerful radio waves, J. Atmos. Sol. Terr. Phys.,

62, 267, 2000.

Yoshik_awa, A., and M. Itonaga, Reflection of shear Alfvdn waves at the ionosphere and the divergent Hall current,

25,824 POKHOTELOV ET AL.' IONOSPHERIC ALFVI•N RESONATOR

Yoshikawa, A., and M. It, onaga, The nature of reflection

and mode conversion of MHD waves in the inductive iono-

sphere: Multistep mode conversion between divergent. and rotational electric fields, J. Geoph:q.q. Re'.q., 105, 10,565,

2000.

Yoshikawa, A., M. Itonaga, S. Fujita, 1t. Nakata, and K. 5\t-

moro, Eigenmode analysis of field line oscillations inter- acting with the ionosphere-atmosphere-solid earth elect. ro-

magnetic coupled system, J. Ge:oph.q:•. Re:s., 10/•, 28,437,

1999.

810 Moscow, Russia. (khru"@uipe-ras.scgis.ru; pokh':•uipe-

ras.scgis.ru; liper@uipe-ras.scgis.ru)

M. Parrot, Laboratoire de Physique et Chimie de l'Environnement, Centre National de la Recherche Scien- tifique, 3A, Avenue de la Recherche Scientifique, F-45071

Orldans, France. (mparrot@odyssee.cm's-orleans.fi')

V. P. Pavlenko, Department of Space and Plasma Physics, Uppsala University, S-75591 Uppsala, Sweden.

( vladimir. pavlenko @rymdfysik.uu. se )

V. Khruschev, O. A. Pokhotelov, S. Senchenkov, Insti-

tute of Physics of the Earth, Bolshaya Gruzinskaya, 10, 123

(Received December 6, 2000; revised May 7', 2001; accepted May 7, 2001.)